TY - JOUR
T1 - Statistical distribution of Fermat quotients
AU - Alexandru, Victor
AU - Cobeli, Cristian
AU - Vâjâitu, Marian
AU - Zaharescu, Alexandru
N1 - Funding Information:
The authors are grateful to Bruce C. Berndt and the referees for their very helpful comments and suggestions.
Publisher Copyright:
© 2022
PY - 2022/8
Y1 - 2022/8
N2 - Let [Formula presented], for 0 ≤ b ≤ p2 − 1 and gcd(b, p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤp is fully covered fast by the quotients qpq with q prime and q ≪ plog2p, as p → ∞ and (ii) The matrix FQMp) passes the pair size correlation test, in which any pair (u, v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.
AB - Let [Formula presented], for 0 ≤ b ≤ p2 − 1 and gcd(b, p) = 1, be the Fermat quotients arranged in the p × (p − 1) Fermat quotient matrix FQM(p). We study the elements of the matrix and prove two results: (i) Under the Generalized Riemann Hypothesis, ℤp is fully covered fast by the quotients qpq with q prime and q ≪ plog2p, as p → ∞ and (ii) The matrix FQMp) passes the pair size correlation test, in which any pair (u, v) of entries in the matrix, which lie apart under any apriori chosen fixed geometric pattern, are in the limit, as p → ∞, with equal probability 1/2 in the size relation u ≤ v or u ≥ v.
KW - Fermat quotient matrix
KW - Fermat quotients
KW - Pair correlation size test
KW - Wieferich primes
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U2 - 10.1016/j.chaos.2022.112335
DO - 10.1016/j.chaos.2022.112335
M3 - Article
AN - SCOPUS:85134605274
SN - 0960-0779
VL - 161
JO - Chaos, Solitons and Fractals
JF - Chaos, Solitons and Fractals
M1 - 112335
ER -